Here \({u_{j}}\)  is the approximation to the point value \(u(x_{j},t)\) 
and \(\hat{f}_{j\pm \frac{1}{2}}\)  are called numerical fluxes which
are Lipschitz continuous in each of its arguments and is consistent
with the physical flux \(\hat{f}(u,....,u)=f(u)\) . The conservation
property is obtained by defining a function \(h(x)\)  implicitly through
the following equation (see Lemma \(2.1\)  of [1]})
\(f(u(x,.))=\frac{1}{\triangle x}_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}h(\xi )d\xi .\) 
